In a conventional two-dimensional Magnetic Resonance Imaging (MRI) scan, a radio-frequency (RF) energy pulse is applied to excite the nuclear spins of the object undergoing scanning. If a slice of the object is selected for scanning, a magnetic field gradient is applied in the direction perpendicular to the slice in conjunction with the RF pulse. As a result, an MRI signal is emitted from the excited slice at the resonant radio frequencies. The magnetic field gradient can be applied in any direction. For simplicity and clarity, the following description assumes that a slice perpendicular to the z-axis is selected for scanning. Thus, to be consistent with the following description, the magnetic field gradient applied with the RF pulse is along the direction of the z-axis.
The emitted MRI signal, denoted as s(k.sub.x, k.sub.y) with k.sub.y set at a constant, represents a one-dimensional spectrum of the slice in two-imensional frequency space, commonly referred to as "k-space". Prior to detection of the MRI signal, a magnetic filed gradient is applied along a transverse direction, or y-axis direction, in order to induce a shift in the phase of the MRI signal in the y-direction of k-space. Additionally, a second RF pulse is commonly applied to refocus the MRI signal, according to a process referred to as generating "echo" of the spins. A magnetic filed gradient along the third orthogonal dimension, or x-axis direction, is thus activated during collection of the MRI signal. The collected MRI signal therefore constitutes a one-dimensional spectrum of the slice along the x-direction, spaced from the x-axis by an amount proportional to the strength and duration of the magnetic field gradient in the y-direction, as shown in k-space by the solid line 100 of FIG. 1.
During a scan sequence, the strength of the y-direction gradient is varied to generate a set of MRI signals having a range of phase shifts, which represent a set of one-dimensional spectra in the x-direction, spaced a plurality of predetermined distances from the x-axis center, as shown by the dashed lines 102 of FIG. 1. The x-direction of k-space is commonly referred to as the "readout" direction, and the y-direction is commonly referred to as the "phase-encoding" direction.
Suppose, for example, that each MRI signal is sampled at a constant interval .DELTA.k.sub.x, along the x-direction in k-space, to provide n.sub.x complex data points. The strength of the y-direction gradient can be incrementally varied, at a constant increment, such that the collected signals are separated by a constant frequency interval .DELTA.k.sub.y along the y-direction in k-space. When a sufficient number, n.sub.y, of MRI signals 100 are collected, the spectra are uniformly distributed. The spatial distribution of the resulting slice f(x, y) can then be reconstructed using a two-dimensional Fourier transform of the k-space MRI signals. That is, ##EQU1##
where s(k.sub.x, k.sub.y) represents the collected MRI signals in k-space, and where f(x,y) represents spatial image data in image-space as described above.
During this process, each collected MRI signal is initially applied to a Fourier transform in the first dimension along the readout direction (x-axis) to generate intermediate results g(x, k.sub.y) as: ##EQU2##
The intermediate results g(x, k.sub.y) are then re-grouped and Fourier transformed in the second dimension along the phase-encoding direction (y-axis) to provide the spatial distribution function f(x, y) of the object: ##EQU3##
where x and y represent discrete positions in the image plane at spatial intervals of .DELTA.x and .DELTA.y, respectively: EQU x=-n.sub.x.DELTA.x/2, -n.sub.x.DELTA.x/2+.DELTA.x, . . . , -.DELTA.x, 0, .DELTA.x, . . . , n.sub.x.DELTA.x/2-2.DELTA.x, n.sub.x.DELTA.x/2-.DELTA.x (4) EQU y=-n.sub.y.DELTA.y/2, -n.sub.y.DELTA.y/2+.DELTA.y, . . . , -.DELTA.y, 0, .DELTA.y, . . . , n.sub.y.DELTA.y/2-2.DELTA.y, n.sub.y.DELTA.y/2-.DELTA.y (5)
In other words, the input data are Fourier transformed row-by-row, and then column-by-column, in k-space, to obtain the spatial data f(x, y). The object image p(x, y), is computed as the magnitude of the complex spatial function f(x, y): EQU p(x, y)=sqrt{f(x, y)f*(x, y)} (6)
where f*(x, y) is the complex conjugate of f(x, y), and "sqrt" represents the square-root function. It should be noted that the MRJ signals s(k.sub.x, k.sub.y) are collected as time-domain data. The data representing the spatial function f(x, y) are corresponding to frequency-domain data, where each point of f(x, y) is associated with certain magnetic resonance frequency. The data g(x, k.sub.y) can be considered as intermediate data with a first dimension in the frequency domain and a second dimension in the time domain.
In the above Equations 1-6, the units are chosen such that the intervals .DELTA.k.sub.x, .DELTA.k.sub.y, .DELTA.x, and .DELTA.y correspond to a value of one. In this scale, the discrete values for k.sub.x, k.sub.y, x, and y become: EQU k.sub.x =-n.sub.x /2, -n.sub.x /2+1, . . . , -1, 0, 1, . . . , n.sub.x /2-2, n.sub.x /2-1; EQU k.sub.y =-n.sub.y /2, -n.sub.y /2+1, . . . , -1, 0, 1, . . . , n.sub.y /2-2, n.sub.y /2-1; EQU x=-n.sub.x /2, -n.sub.x /2+1, . . . , -1, 0, 1, . . . , n.sub.x /2-2, n.sub.x /2-1; and EQU y=-n.sub.y /2, -n.sub.y /2+1, . . . -1, 0, 1, . . . , n.sub.y /2-2, n.sub.y /2-1.
The receiver of a typical MRI scanner is optimized to detect minute MRI signals. In the presence of the RF transmitter, the hyper-sensitive RF receiver inevitably detects a finite, albeit small, level of stray transmitter signal referred to as a "feed-through" signal. This RF feed-through signal results in a corresponding DC offset in the collected base-band MRI signal. Unfortunately, at the receiver, this DC offset is indistinguishable from the true MRI signal emitted from the object at the center of the gradient field. As a consequence, a point artifact having a strong intensity level is generated at the center of the resulting image. The true image intensity at the center is thus completely obscured and inseparable from the point artifact. To complicate matters, the amount of RF feed-through does not necessarily remain constant during a scan. As a result, the DC offset may drift slightly from one phase-encoded signal to another. Consequently, the point artifact 106 spreads out along the direction of the y-axis and thus becomes a line artifact 108 peaking at the center 106 of image space as depicted in FIG. 2. The length l of the line artifact, in other words the number of image pixels affected by the DC offset, depends on the stability of the RF system. For a well-designed system, the length is limited to several pixels.
In addition to RF feed-through, the output of the RF mixer responsible for generating the base-band signal, as well as analog-to-digital converters in the receiver data channels, can also contribute to DC offset. DC offset levels generated by the mixer and the analog-to-digital converters are generally at a much lower magnitude than those of the RF feed-through. The amount of DC offset in each MRI signal is the combined result of these multiple sources.
The expanse and intensity of the point or line artifact are complicated by MRI signal processing. For practical reasons, the collected signal is commonly limited, for example truncated, to a finite sampling length. Such truncation introduces a ripple artifact on the image. In some scans, it is desirable to apply a technique referred to as a "windowing" function along the readout direction (x-axis) of k-space to reduce the truncation effect in that direction. Such a windowing function in k-space is equivalent to a low-pass filtration in image space. As result of this filtering, the line artifact 108 becomes broadened to include multiple lines 108A, 108B, 108C in the resulting image along the x-axis direction, as illustrated in FIG. 3.
It is often even more desirable to apply a second windowing function along the phase-encoding direction (y-axis) of k-space, since the truncation effect in that direction is usually more severe. As a result of this second windowing function along the y-axis, the line artifact 108A is elongated. Furthermore, to reduce scan time, it is common practice to replace collected data with null MRI signals at opposite ends of k-space along the phase-encoding direction, where the MRI signals are often diminished to very low amplitudes, in a process referred to as "zero filling". The truncation effect is worsened by the practice of zero filling, and, as a result, the length of the resulting line artifact 108A in image space is further extended.
The degree of DC offset due to RF feed-through can be somewhat mitigated through proper RF design, system construction and calibration. However, it is practically impossible to reduce the DC offset to a negligible level by these means. In one well-known approach for mitigating DC offset, the MRI signals are acquired twice, or four times, per phase encoding, with the RF pulses applied in alternating phases. The resulting MRI signals in each phase encoding are added and the DC offsets are substantially canceled. However, because the RF feed-through level is continuously drifting, the DC offset is not completely canceled, and the reminiscent artifact is still too large to ignore. Furthermore, this averaging technique can result in a prohibitively long scan time.
Another approach involves discarding the image data affected by the line artifact. For example, the image data along the center line of the image containing the line artifact 108A of FIGS. 2 and 3, are discarded and replaced by values interpolated from the pixels of adjacent lines. Although the interpolation technique is relatively simple and results in a final image that is artifact-free, removal of the artifact in this manner carries with it a setback. Firstly, the artifact occurs in the central region of the image, which is often the region of greatest interest to the observer. The interpolation thus reduces the resolution of the section of pixels superimposed with the artifact at the most interesting region of the image. Secondly, if the phase-encoding, windowing, and/or zero filling techniques are employed by the system, the resulting line artifact 108A is extended as shown in FIG. 3, and thus, virtually the entire set of pixels comprising the central line of the image along the y-axis are interpolated. More seriously, if a windowing function along the readout direction, or x-direction in k-space, is employed, then multiple lines of pixels 108B, 108C are affected by the interpolation, further reducing system resolution.